THE DETERMINATE MACHINE 3/7 



Ex. 19: (Continued.) The system is next changed so that its transformation 

 becomes .v' = j(x + y), y' — j(x — y) + 100. It starts with wages and 

 prices both at 110. Calculate what will happen over the next ten years. 



Ex. 20: (Continued.) Draw an ordinary graph to show how prices and wages 

 will change. 



Ex. 21: Compare the graphs of Exs. 18 and 20. How would the distinction 

 be described in the vocabulary of economics? 



Ex. 22: If the system of Ex. 19 were suddenly disturbed so that wages fell to 

 80 and prices rose to 120, and then left undisturbed, what would happen 

 over the next ten years? (Hint: use (80,120) as operand.) 



Ex. 23 : (Continued.) Draw an ordinary graph to show how wages and prices 

 would change after the disturbance. 



Ex. 24: Is transformation Tone-one between the vectors (xi, ^2) and the vectors 



(xi',X2V 



J. fx\' = 2.V1 + X2 

 \X2 = A"! + X2 



(Hint: If (.vi,a-2) is given, is (.vi',.Y2') uniquely determined? And vice 

 versa ?) 



*Ex. 25: Draw the kinematic graph of the 9-state system whose components 

 are residues: 



x' = X + y 



How many basins has it? 



3/7. (This section may be omitted.) The previous section is of 

 fundamental importance, for it is an introduction to the methods of 

 mathematical physics, as they are applied to dynamic systems. The 

 reader is therefore strongly advised to work through all the exercises, 

 for only in this way can a real grasp of the principles be obtained. 

 If he has done this, he will be better equipped to appreciate the 

 meaning of this section, which summarises the method. 



The physicist starts by naming his variables — Xj, X2, . . . x„. The 

 basic equations of the transformation can then always be obtained 

 by the following fundamental method: — 



(l)Take the first variable, x^, and consider what state it will 

 change to next. If it changes by finite steps the next state will be 

 Xi', if continuously the next state will be .Vj + ^Vj. (In the latter 

 case he may, equivalently, consider the value of dxjdt.) 



(2) Use what is known about the system, and the laws of physics, 

 to express the value of .Yj', or clxjdt (i.e. what x^ will be) in terms 

 of the values that .Yj, . . ., x„ (and any other necessary factors) have 

 now. In this way some equation such as 



X,' = 2a.Yj — .Y3 or dxjch = 4k sin X3 



is obtained. 



35 



